Usenet Newsgroups : rec.puzzles Hall of Fame

The rec.puzzles Hall of Fame is a compilation of over 500 of the most popular puzzles that have been posted and discussed in the rec.puzzles newsgroup. In most cases a detailed solution has been provided.

A person walks up a hill starting at 8 pm and arrives at the top at midnight.
The next day the same person walks down the hill following the same path, again
starting at 8 pm and arriving at midnight. Is there necessarily a point on
the path where the person is at the same time on both days?